Theorem 3netr3g 2391

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)

Hypotheses

Ref	Expression
3netr3g.1	|- ( ph -> A =/= B )
3netr3g.2	|- A = C
3netr3g.3	|- B = D

Assertion

Ref	Expression
3netr3g	|- ( ph -> C =/= D )

Proof of Theorem 3netr3g

Step	Hyp	Ref	Expression
1		3netr3g.1	. 2 |- ( ph -> A =/= B )
2		3netr3g.2	. . 3 |- A = C
3		3netr3g.3	. . 3 |- B = D
4	2, 3	neeq12i 2374	. 2 |- ( A =/= B <-> C =/= D )
5	1, 4	sylib 122	1 |- ( ph -> C =/= D )

Syntax hints:   -> wi 4   = wceq 1363   =/= wne 2357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-cleq 2180  df-ne 2358

This theorem is referenced by: (None)